C₂×#2735974 #72325508

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

direct product

C₂×#2735974

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

56615 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

(4,5)(11,12,15,16)(13,14,17,18)(19,20,22,23)(21,24,26,25);(3,4)(5,6)(8,10)(20,23)(24,25);(4,5)(11,13,15,17)(12,14,16,18)(19,21,22,26)(20,24,23,25);(4,5)(7,8,9,10)(11,13)(12,14)(15,17)(16,18)(19,20)(21,24)(22,23)(25,26);(1,2);(3,5,6,4)(11,14)(12,13)(15,18)(16,17)(19,21)(20,25)(22,26)(23,24)

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232188, C₂×#72232123, C₂×#72232134
maximal normal subgroup of	C₂×#72232188, C₂×#72232123, C₂×#72232134

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((4,5)(11,12,15,16)(13,14,17,18)(19,20,22,23)(21,24,26,25), (3,4)(5,6)(8,10)(20,23)(24,25), (4,5)(11,13,15,17)(12,14,16,18)(19,21,22,26)(20,24,23,25), (4,5)(7,8,9,10)(11,13)(12,14)(15,17)(16,18)(19,20)(21,24)(22,23)(25,26), (1,2), (3,5,6,4)(11,14)(12,13)(15,18)(16,17)(19,21)(20,25)(22,26)(23,24));

C2×#2735974 #72325508